is defined on the interface (the displacement variable \(\mathbf {X}(s,t)\) , for example, is defined on all the nodes). The solid deformation governed by Equation (REF ) is subsequently solved by such a finite difference methodology in a strong form [1]}, [2]}, [3]}. That is, for an arbitrary variable, the central, downwind and upwind difference approximations to the first-order derivatives, are given by
\({\left\lbrace \begin{array}{ll}D^0_s \mathbf {X} & =(\mathbf {X}(s+\Delta s/2)-\mathbf {X}(s-\Delta s/2))/\Delta s,\\D^+_s \mathbf {X} & =(\mathbf {X}(s+\Delta s)-\mathbf {X}(s))/\Delta s,\\D^-_s \mathbf {X} & =(\mathbf {X}(s)-\mathbf {X}(s-\Delta s))/\Delta s,\end{array}\right.}\) 
